## Quantification of the fraction poorly deformable red blood cells using ektacytometry

Optics Express, Vol. 18, Issue 13, pp. 14173-14182 (2010)

http://dx.doi.org/10.1364/OE.18.014173

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### Abstract

We describe a method to obtain the fraction of poorly deformable red blood cells in a blood sample from the intensity pattern in an ektacytometer. In an ektacytometer red blood cells are transformed into ellipsoids by a shear flow between two transparent cylinders. The intensity pattern, due to a laser beam that is sent through the suspension, is projected on a screen. When measuring a healthy red blood cell population iso-intensity curves are ellipses with an axial ratio equal to that of the average red blood cell. In contrast poorly deformable cells result in circular iso-intensity curves. In this study we show that for mixtures of deformable and poorly deformable red blood cells, iso-intensity curves in the composite intensity pattern are neither elliptical nor circular but obtain cross-like shapes. We propose a method to obtain the fraction of poorly deformable red blood cells from those intensity patterns. Experiments with mixtures of poorly deformable and deformable red blood cells validate the method and demonstrate its accuracy. In a clinical setting our approach is potentially of great value for the detection of the fraction of sickle cells in blood samples of patients with sickle cell disease or to find a measure for the parasitemia in patients infected with malaria.

© 2010 OSA

## 1. Introduction

2. J. G. G. Dobbe, M. R. Hardeman, G. J. Streekstra, J. Strackee, C. Ince, and C. A. Grimbergen, “Analyzing red blood cell-deformability distributions,” Blood Cells Mol. Dis. **28**(3), 373–384 (2002). [CrossRef] [PubMed]

4. R. M. Johnson, C. J. Féo, M. Nossal, and I. Dobo, “Evaluation of covalent antisickling compounds by PO_{2} scan ektacytometry,” Blood **66**(2), 432–438 (1985). [PubMed]

5. S. Chien, J. Dormandy, E. Ernst, and A. Matrai, *Clinical Hemorheology* (Martinus Nijhoff publishers, Boston, 1987), p. 238. [PubMed]

9. J. G. G. Dobbe, G. J. Streekstra, M. R. Hardeman, C. Ince, and C. A. Grimbergen, “Measurement of the distribution of red blood cell deformability using an automated rheoscope,” Cytometry **50**(6), 313–325 (2002). [CrossRef] [PubMed]

10. G. J. Streekstra, A. G. Hoekstra, and R. M. Heethaar, “Anomalous diffraction by arbitrarily oriented ellipsoids: applications in ektacytometry,” Appl. Opt. **33**(31), 7288–7296 (1994). [CrossRef] [PubMed]

10. G. J. Streekstra, A. G. Hoekstra, and R. M. Heethaar, “Anomalous diffraction by arbitrarily oriented ellipsoids: applications in ektacytometry,” Appl. Opt. **33**(31), 7288–7296 (1994). [CrossRef] [PubMed]

*l*and a short axis

*s*of the ellipse. In routine measurements, a deformation index

*DI*, defined by (

*l-s*)/(

*l + s*), is measured at different angular velocities of the outer cylinder.

*DI*is plotted against the calculated shear stress in the suspension and represents the deformability of the red cells under consideration. It varies between 0 for low shear stresses to as much as a maximum value of approximately 0.6 at high shear stress (30 Pa).

9. J. G. G. Dobbe, G. J. Streekstra, M. R. Hardeman, C. Ince, and C. A. Grimbergen, “Measurement of the distribution of red blood cell deformability using an automated rheoscope,” Cytometry **50**(6), 313–325 (2002). [CrossRef] [PubMed]

12. M. Bessis, N. Mohandas, and C. Feo, “Automated ektacytometry: a new method of measuring red cell deformability and red cell indices,” Blood Cells **6**(3), 315–327 (1980). [PubMed]

19. G. J. Streekstra, A. G. Hoekstra, E. J. Nijhof, and R. M. Heethaar, “Light scattering by red blood cells in ektacytometry: Fraunhofer versus anomalous diffraction,” Appl. Opt. **32**, 2266–2272 (1993). [CrossRef] [PubMed]

## 2. Isointensity curves of a mixture of oblate and prolate spheroids

19. G. J. Streekstra, A. G. Hoekstra, E. J. Nijhof, and R. M. Heethaar, “Light scattering by red blood cells in ektacytometry: Fraunhofer versus anomalous diffraction,” Appl. Opt. **32**, 2266–2272 (1993). [CrossRef] [PubMed]

*a*>

*b*>

*c*, situated in the origin of a Cartesian coordinate system and

*a*,

*b*and

*c*oriented along the

*x*,

*y*and

*z*axes respectively (Fig. 1 ). The particle is illuminated by a plane wave traveling in the

*z*-direction. In the anomalous diffraction approximation the intensity

*I*at a point (

_{A}*x*,

*y*,

*z*) far from the particle is given by [14

14. M. Bessis, C. Feo, and E. Jones, “Quantitation of red cell deformability during progressive deoxygenation and oxygenation in sickling disorders (the use of an automated Ektacytometer),” Blood Cells **8**(1), 17–28 (1982). [PubMed]

*I*

_{o}denotes the intensity of the incident wave,

*J*(

_{o}*u*) is the zeroth-order Bessel function of

*u*,

*k*is the magnitude of the wave vector of the light in the medium surrounding the particle,

*q*is the axial ratio

*a*/

*b*of the ellipsoid and α is the size parameter for an ellipsoidal particle.

*v*build up curves of equal intensity. On a screen perpendicular to the direction of the incident light these isointensity curves are ellipses. In a red cell suspension with

*uniform*deformation of all the cells within the population, the isointensity curves in the intensity pattern are elliptical with an axial ratio

*q*equal to the axial ratio

_{p}*q*of the cells. Furthermore, the axial ratios of the isointensity curves are independent of the intensity level of these curves. If the cell population consists of a mixture of deformable and poorly deformable cells, the relation between cell shape and intensity pattern is more complicated. To calculate the intensity patterns of such mixtures, we used the fact that in a diluted cell suspension as present in the ektacytometer, the intensity at any point on the screen is the sum of the intensities due to the light scattered by the individual cells [20].

*q*= 4.7) and oblate spheroids (

*q*= 1) representing the deformable and poorly deformable red blood cells. The isointensity curves 1, 2 and 3 in Fig. 2 indicate intensity levels I(0)/2, I(0)/4 and I(0)/10.

*q*from the pattern. However, the value of

_{p}*q*dependends on the intensity level of the isointensity curves. The corresponding values of

_{p}*q*are 1.9, 2.2 and 2.4.

_{p}## 3. Determination of the fraction poorly deformable cells

*q*

_{1}and

*q*

_{2}to these subpopulations [10

10. G. J. Streekstra, A. G. Hoekstra, and R. M. Heethaar, “Anomalous diffraction by arbitrarily oriented ellipsoids: applications in ektacytometry,” Appl. Opt. **33**(31), 7288–7296 (1994). [CrossRef] [PubMed]

*I*(

*x,y,q*

_{1}) and

*I*(

*x,y,q*

_{2}) represent the intensities of the light scattered by cells with axial ratios

*q*

_{1}and

*q*

_{2}. In Eq. (2),

*n*

_{1}and

*n*

_{2}are the numbers of cells with axial ratios

*q*

_{1}and

*q*

_{2}, respectively.

19. G. J. Streekstra, A. G. Hoekstra, E. J. Nijhof, and R. M. Heethaar, “Light scattering by red blood cells in ektacytometry: Fraunhofer versus anomalous diffraction,” Appl. Opt. **32**, 2266–2272 (1993). [CrossRef] [PubMed]

*I*is measured at many points (

_{sc}*x,y*) on a screen. Measurement of the intensity at

*m*different positions on the screen results in

*m*equations like Eq. (2). These equations can be written as one matrix equationIn this matrix equation,

**I**is a vector of length

_{meas}*m*containing the measured intensities

*I*(

_{sc}*x,y*) of the composite intensity pattern,

**I**is a (

_{mat}*m*× 2) matrix and

**n**is a vector with two elements

*n*

_{1}and

*n*

_{2}which are the numbers of cells as in Eq. (2). The rows of

**I**represent the intensities at the

_{mat}*m*positions on the screen for the single cells with axial ratios

*q*

_{1}and

*q*

_{2}. In this way

**I**is the matrix that contains the intensity patterns of the two elementary particles building up the composite intensity patterns.

_{mat}*n1, n2*) with axial ratios

*q*

_{1}and

*q*

_{2}that go with a certain measurement vector

**I**we must invert Eq. (3) to obtain

_{meas}*n*

_{1}and

*n*

_{2}. This inverse relation is given bywhere

**I**is the (2 ×

_{mat}^{−1}*m*) pseudoinverse of

**I**. For the inversion of

_{mat}**I**we use Singular Value Decomposition [16,18].If

_{mat}*n*

_{1}and

*n*

_{2}represent the numbers of poorly deformable and deformable red blood cells in the population, respectively, the fraction of poorly deformable cells

*n*is calculated by Equation (4) produces a solution vector

_{u}**n**that is the best approximation in the least-squares sense [16].

## 4. Experiments

^{4}cells/μl.

*q*of approximately 3. The isointensity curves corresponding to the poorly deformable cells are circular (

*q»*1). The composite intensity patterns due to mixtures of deformable and poorly deformable cells are shown in Fig. 3 (bottom row) and Fig. 4 . In all mixtures the shape of the isointensity curves depends on the intensity level of the curves. This corresponds to the calculations that we performed on mixtures of prolate and oblate ellipsoids.

**I**and a vector

_{mat}**I**had to be obtained from the measurements. For the two columns of

_{meas}**I**, the measured intensity patterns of the poorly deformable (

_{mat}*q*1) and the deformable (

_{1}»*q*3.0) cells were used. To minimize the errors produced by fluctuations in the local red cell concentrations during the measurements, the mean of 10 images were used for each element of

_{2}»**I**. Both columns contained

_{mat}*m*= 45 × 45 elements obtained from the relevant part of the intensity pattern. The intensities of the composite intensity patterns were obtained from the same part of the intensity pattern and collected in the vector

**I**.

_{meas}*n*

_{1}and

*n*

_{2}and the fraction of poorly deformable cells

*n*. Figure 5 shows the value of

_{u}*n*(

_{u}*measured*) obtained from the composite intensity patterns for five prepared mixtures of poorly deformable and deformable red blood cells:

*n*(

_{u}*sample*) = 0.0, 0.25, 0.5, 0.75 and 1.00 respectively. For every mixture, 10 images were scanned and the corresponding values of

*n*were estimated from each image.

_{u}*n*(

_{u}*measured*) and the associated standard deviation are plotted in the figure. The figure shows that the actual fraction of poorly deformable cells represented by the dashed lines is very close to the estimated values. Furthermore, the actual fractions lie within the standard deviation of the measured

*n*.

_{u}## 5. Discussion

*DI*, which is based on elliptical isointensity curves in the intensity pattern, is not the correct parameter to use in this situation. Therefore we presented an alternative method to analyze these intensity patterns which gives accurate quantitative information about the fraction poorly deformable cells in a red cell population.

*n*obtained by our method and the

_{u}*n*of the prepared samples. The standard deviation in the measured

_{u}*n*never exceeds 3%. Apparently, the intensity pattern contains sufficient information in order to allow discrimination between the different mixtures of deformable and poorly deformable cells.

_{u}**I**that contains one column with the intensity pattern of a suspension with only sickled cells and a second row containing the intensity pattern with only deformable cells, it should be possible to obtain the amount of sickled cells in blood samples of sickle cell patients. Once this technique appears to be applicable for sickle cells, a variety of experiments are possible to study the sickling behavior of cells from patients with sickle cell disease. In experiments with blood samples containing sickle cells using the standard ektacytometer setup [14

_{mat}14. M. Bessis, C. Feo, and E. Jones, “Quantitation of red cell deformability during progressive deoxygenation and oxygenation in sickling disorders (the use of an automated Ektacytometer),” Blood Cells **8**(1), 17–28 (1982). [PubMed]

*DI*is observed by changing the oxygen tension (pO

_{2}) and pH. With our method we expect that it will be possible to study the dependencies on pO

_{2}and pH with the actual fraction of sickled cells (

*n*) as a parameter instead of measuring the less appropriate

_{u}*DI*.

## References and links

1. | M. Bessis and N. Mohandas, “Laser Diffraction Patterns of Sickle Cells in Fluid Shear Fields,” Blood Cells |

2. | J. G. G. Dobbe, M. R. Hardeman, G. J. Streekstra, J. Strackee, C. Ince, and C. A. Grimbergen, “Analyzing red blood cell-deformability distributions,” Blood Cells Mol. Dis. |

3. | Y. C. Fung, |

4. | R. M. Johnson, C. J. Féo, M. Nossal, and I. Dobo, “Evaluation of covalent antisickling compounds by PO |

5. | S. Chien, J. Dormandy, E. Ernst, and A. Matrai, |

6. | T. Fischer and H. Schmidt Schönbein, “Tank Tread Motion of red cell membranes in viscometric flow: behavior of intracellular and extracellular markers (with Film),” Blood Cells |

7. | M. Bessis and N. Mohandas, “A Diffractometric Method for the Measurement of Cellular Deformability,” Blood Cells |

8. | M. R. Hardeman, P. T. Goedhart, J. G. G. Dobbe, and K. P. Lettinga, “Laser-assisted Optical Rotational Analyser (LORCA); A new instrument for measurement of various structural hemorheological parameters,” Clin. Hemorheol. |

9. | J. G. G. Dobbe, G. J. Streekstra, M. R. Hardeman, C. Ince, and C. A. Grimbergen, “Measurement of the distribution of red blood cell deformability using an automated rheoscope,” Cytometry |

10. | G. J. Streekstra, A. G. Hoekstra, and R. M. Heethaar, “Anomalous diffraction by arbitrarily oriented ellipsoids: applications in ektacytometry,” Appl. Opt. |

11. | C. Allard, N. Mohandas, and M. Bessis, “Red Cell Deformability Changes in Hemolytic Anemias Estimated by Diffractometric Methods (Ektacytometry),” Blood Cells |

12. | M. Bessis, N. Mohandas, and C. Feo, “Automated ektacytometry: a new method of measuring red cell deformability and red cell indices,” Blood Cells |

13. | J. Plasek and T. Marik, “Determination of undeformable erythrocytes in blood samples using laser light scattering,” Appl. Opt. |

14. | M. Bessis, C. Feo, and E. Jones, “Quantitation of red cell deformability during progressive deoxygenation and oxygenation in sickling disorders (the use of an automated Ektacytometer),” Blood Cells |

15. | D. J. Abraham, A. S. Mehanna, F. C. Wireko, J. Whitney, R. P. Thomas, and E. P. Orringer, “Vanillin, a potential agent for the treatment of sickle cell anemia,” Blood |

16. | L. Lawson, and J. Hanson, |

17. | S. Twomey, |

18. | W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, |

19. | G. J. Streekstra, A. G. Hoekstra, E. J. Nijhof, and R. M. Heethaar, “Light scattering by red blood cells in ektacytometry: Fraunhofer versus anomalous diffraction,” Appl. Opt. |

20. | H. C. van de Hulst, |

21. | M. R. Hardeman, R. M. Bauersachs, and H. J. Meiselman, “RBC Laser diffractometry and RBC Aggregometry with a rotational viscometer: comparison with rheoscope and Myrenne Aggregometer,” Clin. Hemorheol. |

**OCIS Codes**

(290.5850) Scattering : Scattering, particles

**ToC Category:**

Medical Optics and Biotechnology

**History**

Original Manuscript: April 28, 2010

Manuscript Accepted: June 10, 2010

Published: June 16, 2010

**Virtual Issues**

Vol. 5, Iss. 11 *Virtual Journal for Biomedical Optics*

**Citation**

G. J. Streekstra, J. G. G. Dobbe, and A. G. Hoekstra, "Quantification of the fraction poorly deformable red blood cells using ektacytometry," Opt. Express **18**, 14173-14182 (2010)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-18-13-14173

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### References

- M. Bessis and N. Mohandas, “Laser Diffraction Patterns of Sickle Cells in Fluid Shear Fields,” Blood Cells 3, 229–239 (1977).
- J. G. G. Dobbe, M. R. Hardeman, G. J. Streekstra, J. Strackee, C. Ince, and C. A. Grimbergen, “Analyzing red blood cell-deformability distributions,” Blood Cells Mol. Dis. 28(3), 373–384 (2002). [CrossRef] [PubMed]
- Y. C. Fung, Biomechanics, (Springer Verlag, New York, 1984).
- R. M. Johnson, C. J. Féo, M. Nossal, and I. Dobo, “Evaluation of covalent antisickling compounds by PO2 scan ektacytometry,” Blood 66(2), 432–438 (1985). [PubMed]
- S. Chien, J. Dormandy, E. Ernst, and A. Matrai, Clinical Hemorheology (Martinus Nijhoff publishers, Boston, 1987), p. 238. [PubMed]
- T. Fischer and H. Schmidt Schönbein, “Tank Tread Motion of red cell membranes in viscometric flow: behavior of intracellular and extracellular markers (with Film),” Blood Cells 3, 351–365 (1977).
- M. Bessis and N. Mohandas, “A Diffractometric Method for the Measurement of Cellular Deformability,” Blood Cells 1, 307–313 (1975).
- M. R. Hardeman, P. T. Goedhart, J. G. G. Dobbe, and K. P. Lettinga, “Laser-assisted Optical Rotational Analyser (LORCA); A new instrument for measurement of various structural hemorheological parameters,” Clin. Hemorheol. 14(4), 605–619 (1994).
- J. G. G. Dobbe, G. J. Streekstra, M. R. Hardeman, C. Ince, and C. A. Grimbergen, “Measurement of the distribution of red blood cell deformability using an automated rheoscope,” Cytometry 50(6), 313–325 (2002). [CrossRef] [PubMed]
- G. J. Streekstra, A. G. Hoekstra, and R. M. Heethaar, “Anomalous diffraction by arbitrarily oriented ellipsoids: applications in ektacytometry,” Appl. Opt. 33(31), 7288–7296 (1994). [CrossRef] [PubMed]
- C. Allard, N. Mohandas, and M. Bessis, “Red Cell Deformability Changes in Hemolytic Anemias Estimated by Diffractometric Methods (Ektacytometry),” Blood Cells 3, 209–221 (1977).
- M. Bessis, N. Mohandas, and C. Feo, “Automated ektacytometry: a new method of measuring red cell deformability and red cell indices,” Blood Cells 6(3), 315–327 (1980). [PubMed]
- J. Plasek and T. Marik, “Determination of undeformable erythrocytes in blood samples using laser light scattering,” Appl. Opt. 21(23), 4335–4338 (1982). [CrossRef] [PubMed]
- M. Bessis, C. Feo, and E. Jones, “Quantitation of red cell deformability during progressive deoxygenation and oxygenation in sickling disorders (the use of an automated Ektacytometer),” Blood Cells 8(1), 17–28 (1982). [PubMed]
- D. J. Abraham, A. S. Mehanna, F. C. Wireko, J. Whitney, R. P. Thomas, and E. P. Orringer, “Vanillin, a potential agent for the treatment of sickle cell anemia,” Blood 77(6), 1334–1341 (1991). [PubMed]
- L. Lawson, and J. Hanson, Solving Least Squares Problems, (Prentice-Hall, Englewood Cliffs, N.J., 1974), pp. 18.
- S. Twomey, Introduction to the mathematics of inversion in remote sensing and indirect measurements, (Elsevier Scientific Publishing Company, Amsterdam, 1977), pp. 115.
- W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C, (Cambridge University Press, Cambridge, 1988), pp. 528.
- G. J. Streekstra, A. G. Hoekstra, E. J. Nijhof, and R. M. Heethaar, “Light scattering by red blood cells in ektacytometry: Fraunhofer versus anomalous diffraction,” Appl. Opt. 32, 2266–2272 (1993). [CrossRef] [PubMed]
- H. C. van de Hulst, Light Scattering by Small Particles, (Wiley, New York, 1957), pp. 3.
- M. R. Hardeman, R. M. Bauersachs, and H. J. Meiselman, “RBC Laser diffractometry and RBC Aggregometry with a rotational viscometer: comparison with rheoscope and Myrenne Aggregometer,” Clin. Hemorheol. 8, 581–593 (1988).

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